If the kinetic energy of $O_2$ gas is $4.0 \ kJ \ mol^{-1}$,its $RMS$ speed in $cm \ s^{-1}$ is

By what factor does the average velocity of a gaseous molecule increase when the temperature (in Kelvin) is doubled?

In deriving the kinetic gas equation,use is made of the root mean square velocity of the molecules because it is

Which of the following statement$(s)$ is (are) correct regarding the root mean square speed $(U_{rms})$ and average translational kinetic energy $(\varepsilon_{av})$ of a molecule in a gas at equilibrium?
$(1)$ $U_{rms}$ is doubled when its temperature is increased four times
$(2)$ $\varepsilon_{av}$ at a given temperature does not depend on its molecular mass
$(3)$ $U_{rms}$ is inversely proportional to the square root of its molecular mass
$(4)$ $\varepsilon_{av}$ is doubled when its temperature is increased four times

Give the formulas for the following:
$(i)$ Average speed of molecules $(u_{av})$
$(ii)$ Most probable speed $(u_{mp})$
$(iii)$ Root mean square speed $(u_{rms})$

$\alpha, v$ and $u$ represent the most probable velocity, average velocity, and root mean square velocity, respectively, of a gas at a particular temperature. The correct order among the following is: